The way that particular drop down works I don't think it's possible to remove the options you aren't using. I'd probably suggest adding this as an Idea for product mgt to consider.
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Ivanti Support provided me with a solution that works...
In Configure Application go to the Roles and permissions and find the role you want to hide the knowledge types for. Then go to object permissions and find the knowledge type to hide. Then uncheck the add, view, edit and delete permissions for the object. See screen shot below. In this example, we are limiting the knowledge search drop-down to only display Knowledge, IssueResolution and Reference as those are the only types that we use.
If anyone does this though, just be sure to remove the objects (that you removed permission from above) from Incident Object Matching (if you use it). I had Object Matching searching ALL knowledge types so when I removed the object permissions above, I got an error in Incident when Incident Object Matching attempted to search the KB. If you don't know already, see below for an example of how you control what objects Objecting Matching searches. Notice how I am only searching Reference and Issue / Resolution now. By default, Incident Object Matching attempts to search all KB objects.
Ahh of course, that makes sense. Thanks for the update.